[Note for bibliographic reference: Melberg, Hans O. (1998), A principal agent
model, www.oocities.org/hmelberg/papers/980319.htm]
[Note: This paper is not proof-read, and HTML makes it a bit difficult to write
"nice" equations]
A principal agent model
What happens when we have lazy, risk-averse and dishonest people
who know more than us?
by Hans O. Melberg
Introduction
This is my third observation on models in economics. This time I want to present a model
which can be used to improve our understanding of relationships involving asymmetric
information. Amongst other things, the model may help us when we want to understand why
the Soviet economy failed to create intensive and sustained economic growth.
The model
The typical example of asymmetric information involves a landlord and his tenant, although
the relationship is generalizable to many situations (such as between doctors and
patients). The essential point is that these relationships are often characterized by
asymmetric information, that is, one person knows more than the other about some important
variable. For instance, the tenant knows how hard he works, but the landlord may only
observe the actual harvest. Since a good harvest is only partially correlated with high
effort (variables other than effort influence the harvest, such as the weather
conditions), the principal cannot infer the effort of the agent directly from the outcome.
Given this situation of asymmetric information, we might ask what kind of salary the
landlord should offer the principal in order to maximize his profit.
Before we ask this question, it is instructive to see what would happen in a model of
symmetric information. We may then compare the social optimality of this contract to the
contract that will emerge under asymmetric information.
Symmetric information
The principal want to mazimize his expected profit, which can be written as:
r = p x - w
r is profit, p is the price of the product, x is the amount produced and w is the wage
paid to the agent.
Assume that the following relationships hold:
x = de + u : the amount produced is a function of (e)ffort and a random variable (u).
Assume, further, that E(u)=0 and Var(u)>0 (but constant).
The agent's expected utility can be written as:
EU(w) = a + bde - hb2 var(u) - ce2/2
Or, in words: The first part (a + bde) means that the agent receives a fixed sum (a)
and a sum depending on his effort (bde), ie: his wage (w) is: w = a + bx. However, the
agents dislikes uncertainty, so a higher variance mans less utility (which explains the hb2
var (u), h is simply a constant which indicates aversion to variations in income).
Moreover, since we are assuming a "lazy" agent, hard work decreases his utility
(ce2/2, where c is a constant measuring his "aversion" to work).
Lastly, assume that the agent can receive û if he decides not to work for his landlord
(e.g. û is the utility from unemployment benefits).
The principal's problem is to maximize profit subject to the constraint that he has to
offer a contract that gives the agent at least the same utility he could get from
unemployment benefits. That is:
a + bde - hb2 var(u) - ce2/2 = û
Or, rearranged we have:
a + bde = û + hb2 var(u) + ce2/2
Now, the principal will, as mentioned, maximize his expected profit:
r = p x - w
We know that expected production is: x=de. We also know that w= a + bde, so:
r = pde - (a + bde)
Now, before we maximize, we substitute the constraint into the function. That is, we
have an expression for (a + bde) which can be substituted into the profit function:
r = pde - û - hb2 var(u) - ce2/2
To find the optimal wage contract (that is the optimal values for a and b), we
differentiate with respect to b:
dr/db = 2hb var (u) = 0
This implies that in the case of symmetric information, b*=0 (since we have assumed
that h and var(u) both are postive constants, b has to be zero to make the whole
expression zero). If b is zero, a has to be large enough to make the agent's utility at
least as high as the utility from unemployment benefits (otherwise the agent cannot make
the agent work for him). This means that we have:
a* = û + ce2/2
Optimal work effort (from the principal's point of view) is: dr/de = pd - ce = 0
Or,
e* = pd/c
Insert this into a* to get:
a* = û + p2d2/c
In short, under symmetric information we have the following three optimality
conditions:
a* = û + p2d2/c
b*=0
e* = pd/c
Interpreted this means that the agent, or the tenant to return to our original example,
will be offered a contract with a specified level of efffort (e*), a fixed wage (a*), and
no incentive to supply more effort (since b*=0).
Why? The answer is that we have assumed that the agent is risk-averse, while the
principal is risk-neutral. Thus, it is profitable for both parties if the principal takes
the gains and losses from variations, while the risk-averse agent receive a steady income.
Asymmetric information
Assume, now, that the agent knows his own effort, but the principal does not know this
effort (he only observes the final outcome). This means that it is difficult to make a
contract in which the effort level is specified since there is not way of proving that the
agent do not supply the agreed effort (by definition effort is unknown to others except
for the agent who, of course, will say that he worked as hard as he could). Thus, when
offering the agent a contract, the principal has to - first - offer him a contract that
the agent is willing to take (i.e. a utility higher than û). Second, the agent has to
consider the incentives of the agent - how much effort the agent will supply for various
wage systems. The first of these may be called the participation constrant, the second can
be called the incentive constraint. Within these constraint the principal will maximize
his profit.
We already know that when we substitute the participation constraint into the profit
function we get:
r = pde - û - hb2 var(u) - ce2/2
Now, to find the agent's optimal level of effort for a given wage system we simply find
dEU(w)/de. Recall that:
EU(w) = a + bde - hb2 var(u) - ce2/2
So, dEU(w)/de = 0 is:
dEU(w)/de = bd - ce = 0
Or,
e* = bd/c
If we substitute this expression into the profit function we have:
r = pd (bd/c) - û - hb2 var(u) - c (bd/c)2/2
To find the optimal wage contract, we take the differential with respect to b:
dEU(w)/db = pd2/c - 2hb var(u) - bd2/c = 0
Hence, solving for b, we have:
b*=pd2 / [2hc var(u) + d2]
Once again, when we know e* and b*, we can find a* by substituting b* and e* into the
participant contraint (remember, a* must be large enough to make the agent work). We find
that:
a* = û + h(b*)2 var (u) + c(e*)2/2 - b* d e*
In short, we have the following optimal values:
a* = û + h(b*)2 var (u) + c(e*)2/2 - b* d e*
b* = pd2 / [2hc var(u) + d2]
e* = bd/c
Comparison
There are at least two interesting differences between the result above and the optimal
values when there was symmetric information. First, when there is asymmetric information
it is profitable to make some of the salary dependent on the outcome (to give the agent an
incentive to work hard) (This follows from the fact that b*>0 in the case of asymmetric
information). Yet, not all the salary is outcome dependent, since the agent is
risk-averse. In short, the optimal contract is the result of a tradeoff between optimal
risk-allocation and optimal incentive mechanisms.
Second, the effort (and, as a consequence, the amount produced) will be lower under
asymmetric information than symmetric information. Compare:
eS* = pd/c
eA* = bd/c
(Subscript A and S indicate the result under asymmetric and symmetric information.)
Under which conditions will the effort supplied under asymmetric information be the
same as the effort supplied under symmetric information? Recall that:
bA*=pd2 / [2hc var(u) + d2]
If we had had bA*=pd2 / d2, then eA* = pd/c
= eS* There are three possibilities:
1. h=0 (no risk aversion)
2. c=0 (no laziness, no aversion to work)
3. var (u) = 0 (no variation in the outcome)
Since these conditions are unlikely to hold, we can conclude that the optimal contract
between the principal and the agent in the case of asymmetric information is not socially
optimal (most people dislike to work, dislike fluctuations in income and in most cases
random variables affect the outcome). If possible both the agent and the principal would
prefer the symmetric information contract.
Conclusion
I have presented a very condensed version of a model which could have included many more
variables and assumptions. Yet, it is interesting to speculate on the wider implications
of the results. For instance, an economic system which eliminates private property rights
will eventually run into many principal-agent problems since they will have to hire
managers (agents) who do not own the factories themselves. In effect, private property
reduces the number and seriousness of principal-agent type relationships since with
private property there is less incentive to shirk (when you own something you are the one
to suffer if you do not supply enough effort). Of course, the argument is more complicated
and principal-agent models alone do not determine whether private property is a good
thing. Moreover, there are many principal-agent relationships under capitalism
(managers-stockowners, is one). Nevertheless, it is impossible to answer all questions,
and we must advence by examening one argument at a time. Given this limited aim the mode
succeeds well in demonstrating the mechanisms and variables that are involved (the degree
of fluctuation, risk aversion, work aversion and so on).
Note
I have made some verbal comments on principal agent relationships in two previous papers.
See Non-utopian Utopians? (A review of Roemer's book "A Future for Socialism")
and Logical Logic (a review of a book by M. Malia). Especially Roemer is interesting since
he explicitly discusses principal-agent relationships in a systematic fashion
References
My two main sources are:
Williams, M. Lecturenotes (on Principal-Agent theory), Oxford University
Exercises in micro-economics, University of Oslo, 1998.
[Note for bibliographic reference: Melberg, Hans O. (1998), A principal agent model, www.oocities.org/hmelberg/papers/980319.htm]